Heat 1 tablespoon olive oil in a 12-inch skillet over medium-high heat. Season both sides of the chicken breasts with salt and ground black pepper.
Add the chicken to the hot skillet and cook until browned, about 3–5 minutes per side (cook longer for thicker breasts). Remove the chicken to a plate and set aside.
Reduce heat to medium and add the remaining 1 tablespoon olive oil to the skillet. Add the diced onion and cook, stirring occasionally, until the onion is softened, about 2–3 minutes.
Add the minced garlic to the onions and cook, stirring, for about 30 seconds or until fragrant.
Pour in the tomato sauce, then stir in the heavy whipping cream, 1/4 cup grated parmesan cheese, and 1 teaspoon Italian seasoning. Stir to combine, season with salt and ground black pepper to taste, and bring the sauce to a gentle simmer over medium heat.
Return the chicken to the skillet, nestling the breasts into the sauce. Spoon some sauce over the tops of the chicken.
Sprinkle the 1/2 cup shredded parmesan cheese evenly over the chicken. Cover the skillet and cook 1–3 minutes, just until the shredded parmesan has melted and the chicken is heated through. If the chicken was not fully cooked when seared, continue simmering (covered) until the internal temperature reaches 165°F.
Remove from heat, garnish with a little extra Italian seasoning if desired, and serve (optionally over pasta).